tetrahedral Siegel Disk Julia map

From: Roger Bagula (tftn_at_earthlink.net)
Date: 11/24/04 

Date: Wed, 24 Nov 2004 20:16:15 GMT

  Siegel disks don't just happen in complex dynamics of quadratics.

You can set this type of "motion" going on other Riemannian surfaces
as Dr. McMullen suggested in his paper on K3 surfaces
using an tetraheral implicit surface and a Salem based irrational number.
In this simulation an Siegel disk is located on a Riemannian
tetraheral surface.
The Julia in question is:
z'=Lamda*z+(z^4-2*Sqrt[3]*I*z^2+1)/4

Clear[x,y,a,b,s,f,g,a0,t]
(*tetrahedral Siegel Disk Julia map*)
(* idea based on McMullen K3 ( tetrahedral) surface Siegel disk dynamics*)
z=x[n-1,t]+I*y[n-1,t]
z4=ComplexExpand[z^4-2*Sqrt[3]*I*z^2+1]
(* Riemannian Tetrahedron polynomial from Elliptic Curves, McKean and Moll,
  p22, Ellipical invariants of Platonic solids*)
(* j[z]=(z^4-2*Sqrt[3]*I*z^2+1)^3/(z^4+2*Sqrt[3]*I*z^2+1) *)
f[n_,t_]=Re[z4]
g[n_,t_]=Im[z4]
gm=N[(1+Sqrt[5])/2];
a=Cos[2*Pi*gm];
b=Sin[2*Pi*gm];
digits=1500;
x[n_,t_]:=x[n,t]=x[n-1,t]*a-y[n-1,t]*b+f[n,t]/4
y[n_,t_]:=y[n,t]=x[n-1,t]*b+y[n-1,t]*a+g[n,t]/4
x[0,t_]:=0.27/(1+t/2);y[0,t_]=0.01/(1+t/2);
a=Flatten[Table[Table[{x[n,t],y[n,t]},{n,0, digits}],{t,1,10}],1];
ListPlot[a, PlotRange->All]
Respectfully, Roger L. Bagula

tftn@earthlink.net, 11759Waterhill Road, Lakeside,Ca 92040-2905,tel: 619-5610814 :
alternative email: rlbtftn@netscape.net
URL : http://home.earthlink.net/~tftn

Siegel_tetrahedron.gif
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